while the Higgs doublet develops its VEV as is customary \(\left\langle H_{u}\right\rangle =\upsilon _{u}\) . Then, using the tensor product of \(D_{4}\) 
irreducible representations (see eqs. (C.2) and (C.3) of Ref. [1]}), we find that the Dirac and Majorana mass matrices have the following
forms
\(m_{D}=\upsilon _{u}Y_{D}=\upsilon _{u}\left(\begin{array}{ccc}y_{1} & 0 & 0 \\0 & y_{2} & 0 \\0 & 0 & y_{2}\end{array}\right) \quad ,\quad m_{M}=\left(\begin{array}{ccc}y_{3}\upsilon _{\rho _{1}} & y_{5}\upsilon _{\digamma } & y_{5}\upsilon _{\digamma }+y_{6}\upsilon _{\Gamma } \\y_{5}\upsilon _{\digamma } & y_{7}\upsilon _{\rho _{2}}-y_{8}\upsilon _{\rho _{3}} & 2y_{4}\upsilon _{\rho _{1}} \\y_{5}\upsilon _{\digamma }+y_{6}\upsilon _{\Gamma } & 2\lambda _{4}\upsilon _{\rho _{1}} & y_{7}\upsilon _{\rho _{2}}+y_{8}\upsilon _{\rho _{3}}\end{array}\right) \) 
